American Institute of Mathematical Sciences

February  2011, 5(1): 137-166. doi: 10.3934/ipi.2011.5.137

A Mumford-Shah level-set approach for the inversion and segmentation of SPECT/CT data

 1 Industrial Mathematics Institute, Johannes Kepler University Linz, Altenbergerstraβe 69, A-4040 Linz, Austria, Austria 2 Institut für Mathematik, Universität Graz, Heinrichstrasse 36, A-8010 Graz, Austria

Received  March 2009 Revised  February 2010 Published  February 2011

This paper presents a level-set based approach for the simultaneous reconstruction and segmentation of the activity as well as the density distribution from tomography data gathered by an integrated SPECT/CT scanner.
Activity and density distributions are modeled as piecewise constant functions. The segmenting contours and the corresponding function values of both the activity and the density distribution are found as minimizers of a Mumford-Shah like functional over the set of admissible contours and -- for fixed contours -- over the spaces of piecewise constant density and activity distributions which may be discontinuous across their corresponding contours. For the latter step a Newton method is used to solve the nonlinear optimality system. Shape sensitivity calculus is used to find a descent direction for the cost functional with respect to the geometrical variables which leads to an update formula for the contours in the level-set framework. A heuristic approach for the insertion of new components for the activity as well as the density function is used. The method is tested for synthetic data with different noise levels.
Citation: Esther Klann, Ronny Ramlau, Wolfgang Ring. A Mumford-Shah level-set approach for the inversion and segmentation of SPECT/CT data. Inverse Problems and Imaging, 2011, 5 (1) : 137-166. doi: 10.3934/ipi.2011.5.137
References:
 [1] G. Aubert and P. Kornprobst, "Mathematical Problems in Image Processing. Partial Differential Equations and the Calculus of Variations, with a Foreword by Olivier Faugeras," Springer-Verlag, New York, 2002. [2] H. B. Ameur, M. Burger and B. Hackl, Level set methods for geometric inverse problems in linear elasticity, Inverse Problems, 20 (2004), 673-696. doi: 10.1088/0266-5611/20/3/003. [3] H. B. Ameur, M. Burger and B. Hackl, Cavity identification in linear elasticity and thermoelasticity, Math. Methods Appl. Sci., 30 (2007), 625-647. doi: 10.1002/mma.772. [4] A. K. Buck, S. Nekolla, S. Ziegler, A. Beer, B. J. Krause, K. Herrmann, K. Scheidhauer, H.-J. Wester, E. J. Rummeny, M. Schwaiger and A. Drzezga, SPECT/CT, The Journal of Nuclear Medicine, 49 (2008), 1305-1319. [5] J. K. Bucsko, SPECT/CT - The future is clear, Radiology Today, 5 (2004). [6] M. Burger, Levenberg-Marquardt level set methods for inverse obstacle problems, Inverse Problems, 20 (2004), 259-282. doi: 10.1088/0266-5611/20/1/016. [7] V. Caselles, F. Catté, T. Coll and F. Dibos, A geometric model for active contours in image processing, Numer. Math., 66 (1993), 1-31. doi: 10.1007/BF01385685. [8] A. Chambolle, Image segmentation by variational methods: Mumford and Shah functional and the discrete approximations, SIAM J. Appl. Math., 55 (1995), 827-863. doi: 10.1137/S0036139993257132. [9] T. F. Chan and L. A. Vese, "Image Segmentation Using Level Sets and the Piecewise Constant Mumford-Shah Model," UCLA CAM Report 00-14, University of California, Los Angeles, 2000. [10] T. F. Chan and L. A. Vese, "A Level Set Algorithm for Minimizing the Mumford-Shah Functional in Image Processing," UCLA CAM Report 00-13, University of California, Los Angeles, 2000. [11] T. F. Chan and L. A. Vese, Active contours without edges, IEEE Trans. Image Processing, 10 (2001), 266-277. doi: 10.1109/83.902291. [12] T. F. Chan and X.-C. Tai, Level set and total variation regularization for elliptic inverse problems with discontinuous coefficients, J. Comput. Phys., 193 (2004), 40-66. doi: 10.1016/j.jcp.2003.08.003. [13] L. D. Cohen and R. Kimmel, Global minimum for active contour models: A minimum path approach, International Journal of Computer Vision, 24 (1997), 57-78. doi: 10.1023/A:1007922224810. [14] D. Delbeke, R. E. Coleman, M. J. Guiberteau, M. L. Brown, H. D. Royal, B. A. Siegel, D. W. Townsend, L. L. Berland, J. A. Parker, G. Zubal and V. Cronin, Procedure Guideline for SPECT/CT Imaging, The Journal of Nuclear Medicine, 47 (2006), 1227-1234. [15] M. C. Delfour and J.-P. Zolésio, "Shapes and Geometries. Analysis, Differential Calculus, and Optimization," Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. [16] V. Dicken, A new approach towards simultaneous activity and attenuation reconstruction in emission tomography, Inverse Probl., 15 (1999), 931-960. doi: 10.1088/0266-5611/15/4/307. [17] O. Dorn, Shape reconstruction in scattering media with voids using a transport model and level sets, Can. Appl. Math. Q., 10 (2002), 239-275. [18] O. Dorn, E. L. Miller and C. M. Rappaport, A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets. Electromagnetic imaging and inversion of the Earth's subsurface, Inverse Problems, 16 (2000), 1119-1156. doi: 10.1088/0266-5611/16/5/303. [19] J.-P. Guillement, F. Jauberteau, L. Kunyansky, R. Novikov and R. Trebossen, On single-photon emission computed tomography imaging based on an exact formula for the nonuniform attenuation correction, Inverse Probl., 18 (2002). [20] M. Hintermüller and W. Ring, "A Level Set Approach for the Solution of a State Constrained Optimal Control Problem," Technical Report 212, Special Research Center on Optimization and Control, University of Graz, Austria, 2001. (to appear in Numerische Mathematik) [21] M. Hintermüller and W. Ring, An inexact Newton-CG-type active contour approach for the minimization of the Mumford-Shah functional, J. Math. Imag. Vis., 20 (2004), 19-42. doi: 10.1023/B:JMIV.0000011317.13643.3a. [22] M. Hintermüller and W. Ring, A second order shape optimization approach for image segmentation, SIAM J. Appl. Math., 64 (2003), 442-467. doi: 10.1137/S0036139902403901. [23] K. Ito, K. Kunisch and Z. Li, Level-set function approach to an inverse interface problem, Inverse Problems, 17 (2001), 1225-1242. doi: 10.1088/0266-5611/17/5/301. [24] S. Jehan-Besson, M. Barlaud and G. Aubert, DREAM$^2$S: Deformable regions driven by an Eulerian accurate minimization method for image and video segmentation, November 2001. [25] S. Jehan-Besson, M. Barlaud and G. Aubert, Video object segmentation using Eulerian region-based active contours, in "International Conference on Computer Vision, Vancouver, July 2001." [26] M. Kass, A. Witkin and D. Terzopoulos, Snakes: Active contour models, Int. J. of Computer Vision, 1 (1987), 321-331. doi: 10.1007/BF00133570. [27] H. Kudo and H. Nakamura, A new appraoch to SPECT attenuation correction without transmission maesurements, Nuclear Science Symposium Conference Record, 2 (2000), 13/58-13/62. [28] L. A. Kunyansky, A new SPECT reconstruction algorithm based on the Novikov explicit inversion formula, Inverse Probl., 17 (2001), 293-306. doi: 10.1088/0266-5611/17/2/309. [29] A. Litman, D. Lesselier and F. Santosa, Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set, Inverse Problems, 14 (1998), 685-706. doi: 10.1088/0266-5611/14/3/018. [30] M. Mancas, B. Gosselin and B. Macq, Segmentation using a region-growing thresholding, in "Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series," (E. R. Dougherty, J. T. Astola and K. O. Egiazarian, eds.), volume 5672 of Presented at the Society of Photo-Optical Instrumentation Engineers (SPIE) Conference, (2005), 388-398. [31] D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., 42 (1989), 577-685. doi: 10.1002/cpa.3160420503. [32] F. Natterer, Inversion of the attenuated Radon transform, Inverse Probl., 17 (2001), 113-119. doi: 10.1088/0266-5611/17/1/309. [33] F. Natterer, "The Mathematics of Computerized Tomography," Volume 32 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. (reprint of the 1986 original) [34] F. Natterer, Determination of tissue attenuation in emission tomography of optically dense media, Inverse Probl., 9 (1993), 731-736. doi: 10.1088/0266-5611/9/6/009. [35] R. G. Novikov, An inversion formula for the attenuated $X$-ray transformation, Ark. Mat., 40 (2002), 145-167. doi: 10.1007/BF02384507. [36] S. J. Osher and R. P. Fedkiw, "Level Set Methods and Dynamic Implicit Surfaces," Springer Verlag, New York, 2002. [37] S. J. Osher and F. Santosa, Level set methods for optimization problems involving geometry and constraints. I, Frequencies of a two-density inhomogeneous drum, J. Comput. Phys., 171 (2001), 272-288. doi: 10.1006/jcph.2001.6789. [38] N. Paragios and R. Deriche, Geodesic active regions: A new paradigm to deal with frame partition problems in computer vision, International Journal of Visual Communication and Image Representation, Special Issue on Partial Differential Equations in Image Processing, Computer Vision and Computer Graphics, 2001. (to appear in 2001) [39] G. N. Ramachandran and A. V. Lakshminarayanan, Three-dimensional reconstruction from radiographs and electron micrographs: Application of convolutions instead of fourier transforms, PNAS, 68 (1971), 2236-2240. doi: 10.1073/pnas.68.9.2236. [40] R. Ramlau, R. Clackdoyle, R. Noo and G. Bal, Accurate attenuation correction in SPECT imaging using optimization of bilinear functions and assuming an unknown spatially-varying attenuation distribution, ZAMM, Z. Angew. Math. Mech., 80 (2000), 613-621. doi: 10.1002/1521-4001(200009)80:9<613::AID-ZAMM613>3.0.CO;2-9. [41] R. Ramlau, TIGRA-An iterative algorithm for regularizing nonlinear ill-posed problems, Inverse Probl., 19 (2003), 433-465. [42] R. Ramlau and W. Ring, A Mumford-Shah level-set approach for the inversion and segmentation of X-ray tomography data, J. Comput. Phys., 221 (2007), 539-557. doi: 10.1016/j.jcp.2006.06.041. [43] F. Santosa, A level-set approach for inverse problems involving obstacles, ESAIM: Control, Optimization and Calculus of Variations, 1 (1996), 17-33. doi: 10.1051/cocv:1996101. [44] G. Sapiro, "Geometric Partial Differential Equations and Image Analysis," Cambridge University Press, Cambridge, 2001. [45] J. A. Sethian, "Level Set Methods and Fast Marching Methods. Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science," Cambridge University Press, Cambridge, second edition, 1999. [46] L. A. Shepp and B. F. Logan, The Fourier reconstruction of a head section, IEEE Trans. Nucl. Sci, (1974), 21-43. [47] J. Sokołowski and J.-P. Zolésio, "Introduction to Shape Optimization. Shape Sensitivity Analysis," Springer-Verlag, Berlin, 1992. [48] J. A. Terry, B. M. W. Tsui, J. R. Perry, J. L. Hendricks and G. T. Gullberg, The design of a mathematical phantom of the upper human torso for use in 3-d spect imaging research, in "Proc. 1990 Fall Meeting Biomed. Eng. Soc., Blacksburg, VA," New York University Press, (1990), 1467-1474. [49] D. Terzopoulos, Deformable models: Classic, topology-adaptive and generalized formulations, in "Geometric Level Set Methods In Imaging, Vision, and Graphics," Springer, New York, (2003), 21-40. doi: 10.1007/0-387-21810-6_2. [50] O. Tretiak and C. Metz, The exponential Radon transform, SIAM J. Appl. Math., 39 (1980), 341-354. doi: 10.1137/0139029. [51] A. Tsai, A. Yezzi and A. S. Willsky, Curve evolution implementation of the Mumford-Shah functional for image segementation, denoising, interpolation, and magnification, IEEE Transactions on Image Processing, 10 (2001), 1169-1186. doi: 10.1109/83.935033. [52] J. Weickert, "Anisotropic Diffusion in Image Processing," European Consortium for Mathematics in Industry, Teubner, Stuttgart, Leipzig, 1998. [53] A. Welch, R. Clack, F. Natterer and G. T. Herman, Toward accurate attenuation correction in SPECT without transmission measurements, IEEE Trans. Med. Imaging, 16 (1997), 532-541. doi: 10.1109/42.640743. [54] M. N. Wernick and J. N. Aarsvold (eds.), "Emission Tomography. The Fundamentals of PET and SPECT," Elsevier Academic Press, 2004. [55] A. D. Zacharopoulos, S. R. Arridge, O. Dorn, V. Kolehmainen and J. Sikora, Three-dimensional reconstruction of shape and piecewise constant region values for optical tomography using spherical harmonic parametrization and a boundary element method, Inverse Probl., 22 (2006), 1509-1532. doi: 10.1088/0266-5611/22/5/001.

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References:
 [1] G. Aubert and P. Kornprobst, "Mathematical Problems in Image Processing. Partial Differential Equations and the Calculus of Variations, with a Foreword by Olivier Faugeras," Springer-Verlag, New York, 2002. [2] H. B. Ameur, M. Burger and B. Hackl, Level set methods for geometric inverse problems in linear elasticity, Inverse Problems, 20 (2004), 673-696. doi: 10.1088/0266-5611/20/3/003. [3] H. B. Ameur, M. Burger and B. Hackl, Cavity identification in linear elasticity and thermoelasticity, Math. Methods Appl. Sci., 30 (2007), 625-647. doi: 10.1002/mma.772. [4] A. K. Buck, S. Nekolla, S. Ziegler, A. Beer, B. J. Krause, K. Herrmann, K. Scheidhauer, H.-J. Wester, E. J. Rummeny, M. Schwaiger and A. Drzezga, SPECT/CT, The Journal of Nuclear Medicine, 49 (2008), 1305-1319. [5] J. K. Bucsko, SPECT/CT - The future is clear, Radiology Today, 5 (2004). [6] M. Burger, Levenberg-Marquardt level set methods for inverse obstacle problems, Inverse Problems, 20 (2004), 259-282. doi: 10.1088/0266-5611/20/1/016. [7] V. Caselles, F. Catté, T. Coll and F. Dibos, A geometric model for active contours in image processing, Numer. Math., 66 (1993), 1-31. doi: 10.1007/BF01385685. [8] A. Chambolle, Image segmentation by variational methods: Mumford and Shah functional and the discrete approximations, SIAM J. Appl. Math., 55 (1995), 827-863. doi: 10.1137/S0036139993257132. [9] T. F. Chan and L. A. Vese, "Image Segmentation Using Level Sets and the Piecewise Constant Mumford-Shah Model," UCLA CAM Report 00-14, University of California, Los Angeles, 2000. [10] T. F. Chan and L. A. Vese, "A Level Set Algorithm for Minimizing the Mumford-Shah Functional in Image Processing," UCLA CAM Report 00-13, University of California, Los Angeles, 2000. [11] T. F. Chan and L. A. Vese, Active contours without edges, IEEE Trans. Image Processing, 10 (2001), 266-277. doi: 10.1109/83.902291. [12] T. F. Chan and X.-C. Tai, Level set and total variation regularization for elliptic inverse problems with discontinuous coefficients, J. Comput. Phys., 193 (2004), 40-66. doi: 10.1016/j.jcp.2003.08.003. [13] L. D. Cohen and R. Kimmel, Global minimum for active contour models: A minimum path approach, International Journal of Computer Vision, 24 (1997), 57-78. doi: 10.1023/A:1007922224810. [14] D. Delbeke, R. E. Coleman, M. J. Guiberteau, M. L. Brown, H. D. Royal, B. A. Siegel, D. W. Townsend, L. L. Berland, J. A. Parker, G. Zubal and V. Cronin, Procedure Guideline for SPECT/CT Imaging, The Journal of Nuclear Medicine, 47 (2006), 1227-1234. [15] M. C. Delfour and J.-P. Zolésio, "Shapes and Geometries. Analysis, Differential Calculus, and Optimization," Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. [16] V. Dicken, A new approach towards simultaneous activity and attenuation reconstruction in emission tomography, Inverse Probl., 15 (1999), 931-960. doi: 10.1088/0266-5611/15/4/307. [17] O. Dorn, Shape reconstruction in scattering media with voids using a transport model and level sets, Can. Appl. Math. Q., 10 (2002), 239-275. [18] O. Dorn, E. L. Miller and C. M. Rappaport, A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets. Electromagnetic imaging and inversion of the Earth's subsurface, Inverse Problems, 16 (2000), 1119-1156. doi: 10.1088/0266-5611/16/5/303. [19] J.-P. Guillement, F. Jauberteau, L. Kunyansky, R. Novikov and R. Trebossen, On single-photon emission computed tomography imaging based on an exact formula for the nonuniform attenuation correction, Inverse Probl., 18 (2002). [20] M. Hintermüller and W. Ring, "A Level Set Approach for the Solution of a State Constrained Optimal Control Problem," Technical Report 212, Special Research Center on Optimization and Control, University of Graz, Austria, 2001. (to appear in Numerische Mathematik) [21] M. Hintermüller and W. Ring, An inexact Newton-CG-type active contour approach for the minimization of the Mumford-Shah functional, J. Math. Imag. Vis., 20 (2004), 19-42. doi: 10.1023/B:JMIV.0000011317.13643.3a. [22] M. Hintermüller and W. Ring, A second order shape optimization approach for image segmentation, SIAM J. Appl. Math., 64 (2003), 442-467. doi: 10.1137/S0036139902403901. [23] K. Ito, K. Kunisch and Z. Li, Level-set function approach to an inverse interface problem, Inverse Problems, 17 (2001), 1225-1242. doi: 10.1088/0266-5611/17/5/301. [24] S. Jehan-Besson, M. Barlaud and G. Aubert, DREAM$^2$S: Deformable regions driven by an Eulerian accurate minimization method for image and video segmentation, November 2001. [25] S. Jehan-Besson, M. Barlaud and G. Aubert, Video object segmentation using Eulerian region-based active contours, in "International Conference on Computer Vision, Vancouver, July 2001." [26] M. Kass, A. Witkin and D. Terzopoulos, Snakes: Active contour models, Int. J. of Computer Vision, 1 (1987), 321-331. doi: 10.1007/BF00133570. [27] H. Kudo and H. Nakamura, A new appraoch to SPECT attenuation correction without transmission maesurements, Nuclear Science Symposium Conference Record, 2 (2000), 13/58-13/62. [28] L. A. Kunyansky, A new SPECT reconstruction algorithm based on the Novikov explicit inversion formula, Inverse Probl., 17 (2001), 293-306. doi: 10.1088/0266-5611/17/2/309. [29] A. Litman, D. Lesselier and F. Santosa, Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set, Inverse Problems, 14 (1998), 685-706. doi: 10.1088/0266-5611/14/3/018. [30] M. Mancas, B. Gosselin and B. Macq, Segmentation using a region-growing thresholding, in "Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series," (E. R. Dougherty, J. T. Astola and K. O. Egiazarian, eds.), volume 5672 of Presented at the Society of Photo-Optical Instrumentation Engineers (SPIE) Conference, (2005), 388-398. [31] D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., 42 (1989), 577-685. doi: 10.1002/cpa.3160420503. [32] F. Natterer, Inversion of the attenuated Radon transform, Inverse Probl., 17 (2001), 113-119. doi: 10.1088/0266-5611/17/1/309. [33] F. Natterer, "The Mathematics of Computerized Tomography," Volume 32 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. (reprint of the 1986 original) [34] F. Natterer, Determination of tissue attenuation in emission tomography of optically dense media, Inverse Probl., 9 (1993), 731-736. doi: 10.1088/0266-5611/9/6/009. [35] R. G. Novikov, An inversion formula for the attenuated $X$-ray transformation, Ark. Mat., 40 (2002), 145-167. doi: 10.1007/BF02384507. [36] S. J. Osher and R. P. Fedkiw, "Level Set Methods and Dynamic Implicit Surfaces," Springer Verlag, New York, 2002. [37] S. J. Osher and F. Santosa, Level set methods for optimization problems involving geometry and constraints. I, Frequencies of a two-density inhomogeneous drum, J. Comput. Phys., 171 (2001), 272-288. doi: 10.1006/jcph.2001.6789. [38] N. Paragios and R. Deriche, Geodesic active regions: A new paradigm to deal with frame partition problems in computer vision, International Journal of Visual Communication and Image Representation, Special Issue on Partial Differential Equations in Image Processing, Computer Vision and Computer Graphics, 2001. (to appear in 2001) [39] G. N. Ramachandran and A. V. Lakshminarayanan, Three-dimensional reconstruction from radiographs and electron micrographs: Application of convolutions instead of fourier transforms, PNAS, 68 (1971), 2236-2240. doi: 10.1073/pnas.68.9.2236. [40] R. Ramlau, R. Clackdoyle, R. Noo and G. Bal, Accurate attenuation correction in SPECT imaging using optimization of bilinear functions and assuming an unknown spatially-varying attenuation distribution, ZAMM, Z. Angew. Math. Mech., 80 (2000), 613-621. doi: 10.1002/1521-4001(200009)80:9<613::AID-ZAMM613>3.0.CO;2-9. [41] R. Ramlau, TIGRA-An iterative algorithm for regularizing nonlinear ill-posed problems, Inverse Probl., 19 (2003), 433-465. [42] R. Ramlau and W. Ring, A Mumford-Shah level-set approach for the inversion and segmentation of X-ray tomography data, J. Comput. Phys., 221 (2007), 539-557. doi: 10.1016/j.jcp.2006.06.041. [43] F. Santosa, A level-set approach for inverse problems involving obstacles, ESAIM: Control, Optimization and Calculus of Variations, 1 (1996), 17-33. doi: 10.1051/cocv:1996101. [44] G. Sapiro, "Geometric Partial Differential Equations and Image Analysis," Cambridge University Press, Cambridge, 2001. [45] J. A. Sethian, "Level Set Methods and Fast Marching Methods. Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science," Cambridge University Press, Cambridge, second edition, 1999. [46] L. A. Shepp and B. F. Logan, The Fourier reconstruction of a head section, IEEE Trans. Nucl. Sci, (1974), 21-43. [47] J. Sokołowski and J.-P. Zolésio, "Introduction to Shape Optimization. Shape Sensitivity Analysis," Springer-Verlag, Berlin, 1992. [48] J. A. Terry, B. M. W. Tsui, J. R. Perry, J. L. Hendricks and G. T. Gullberg, The design of a mathematical phantom of the upper human torso for use in 3-d spect imaging research, in "Proc. 1990 Fall Meeting Biomed. Eng. Soc., Blacksburg, VA," New York University Press, (1990), 1467-1474. [49] D. Terzopoulos, Deformable models: Classic, topology-adaptive and generalized formulations, in "Geometric Level Set Methods In Imaging, Vision, and Graphics," Springer, New York, (2003), 21-40. doi: 10.1007/0-387-21810-6_2. [50] O. Tretiak and C. Metz, The exponential Radon transform, SIAM J. Appl. Math., 39 (1980), 341-354. doi: 10.1137/0139029. [51] A. Tsai, A. Yezzi and A. S. Willsky, Curve evolution implementation of the Mumford-Shah functional for image segementation, denoising, interpolation, and magnification, IEEE Transactions on Image Processing, 10 (2001), 1169-1186. doi: 10.1109/83.935033. [52] J. Weickert, "Anisotropic Diffusion in Image Processing," European Consortium for Mathematics in Industry, Teubner, Stuttgart, Leipzig, 1998. [53] A. Welch, R. Clack, F. Natterer and G. T. Herman, Toward accurate attenuation correction in SPECT without transmission measurements, IEEE Trans. Med. Imaging, 16 (1997), 532-541. doi: 10.1109/42.640743. [54] M. N. Wernick and J. N. Aarsvold (eds.), "Emission Tomography. The Fundamentals of PET and SPECT," Elsevier Academic Press, 2004. [55] A. D. Zacharopoulos, S. R. Arridge, O. Dorn, V. Kolehmainen and J. Sikora, Three-dimensional reconstruction of shape and piecewise constant region values for optical tomography using spherical harmonic parametrization and a boundary element method, Inverse Probl., 22 (2006), 1509-1532. doi: 10.1088/0266-5611/22/5/001.
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